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\topic{Lecture 10 \\Multiple Integral\\ \scriptsize Dirichlet's  Theorem and Liouville's Thoerem
(6 Oct 2009)
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\section{Dirichlet's  Theorem}
\begin{theorem}
If $l, m, n$ are all positive, then the triple integral 
\[\int\int\int_V x^{l-1}y^{m-1}z^{n-1}dadydz = \frac{\Gamma(l)\Gamma(m)\Gamma(n)}{\Gamma(l+m+n+1)}\]
where $V$ is the region $x \geq 0, y \geq 0, z \geq 0 and x+y+z\leq 0$.
\end{theorem}
Remark: This theorem is true for two variable. It is also true for variables more than three.
\begin{example}
Find the mass of an octant of the ellipsoid $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. The density at any point is $\rho=kxyz$.
\end{example}
We have,
\[Mass = \int\int\int_V \rho dV = \int\int\int_V kxyz dxdydz\]
where $V$ is given ellipsoid. Now, we will convert this problem equivalent to dirichlet's problem.

Let $\frac{x^2}{a^2} = X$, $\frac{y^2}{b^2} = Y$, $\frac{z^2}{c^2} = X$, $\Rightarrow X+Y+Z=1$. 

Therefore, $xdx = \frac{a^2 dX}{2}$, $ydy = \frac{b^2 dY}{2}$, $zdz = \frac{c^2 dZ}{2}$.

Thus
\[Mass = \int\int\int_V kxyz dxdydz = \int\int\int_V k\frac{a^2 dX}{2}\frac{b^2 dY}{2}\frac{c^2 dZ}{2} \]
\[ = \frac{ka^2b^2c^2}{8}\int\int\int_V dXdYdZ = \frac{ka^2b^2c^2}{8}\int\int\int_V X^{1-1}Y^{1-1}Z^{1-1} dXdYdZ  \]
\[=\frac{ka^2b^2c^2}{8}\frac{\Gamma(1)\Gamma(1)\Gamma(1)}{\Gamma(1+1+1+1)} = \frac{ka^2b^2c^2}{48}\]
\section{Liouville's Thoerem}
\begin{theorem}
If $l, m, n$ are all positive, then the triple integral 
\[\int\int\int_V f(x+y+z)x^{l-1}y^{m-1}z^{n-1}dadydz = \frac{\Gamma(l)\Gamma(m)\Gamma(n)}{\Gamma(l+m+n)}\int_{h_1}^{h_2}f(u)u^{l+m+n+1}du\]
where $V$ is the region $x \geq 0, y \geq 0, z \geq 0 and h_1 \leq x+y+z \leq h_2$.
\end{theorem}
\begin{example}
Show that
\[\int\int\int \frac{dxdydz}{(x+y+z+1)^3}=\frac{1}{2}\log 2 - \frac{5}{16}\]
the integral being taken throughout the volume bounded by planes $x=0,y=0,z=0, x+y+z=1$
\end{example}
Given
\[\int\int\int \frac{dxdydz}{(x+y+z+1)^3} = \int\int\int \frac{x^{1-1}y^{1-1}z^{1-1}dxdydz}{(x+y+z+1)^3}\]
By Liouville's Thoerem, when $0 \leq x+y+z \leq 1$
\[\int\int\int \frac{x^{1-1}y^{1-1}z^{1-1}dxdydz}{(x+y+z+1)^3} = \frac{\Gamma(1)\Gamma(1)\Gamma(1)}{\Gamma(1+1+1)}\int_0^1 \frac{1}{(u+1)^3}u^{1+1+1-1} du\]
Using partial fraction
\[=\frac{1}{2}\int_0^1 [(u+1) - 2(u+1)^{-2}-(u+1)^{-3}] du\]
\[=\frac{1}{2} [\log(u+1) + 2(u+1) + \frac{1}{2}(u+1)^{-2}]_0^1 du\]
\[=\frac{1}{2}\log 2 - \frac{5}{16}\]
\end{example}
\section*{Problems}
\begin{enumerate}
	\item 
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